On the characteristic polynomial of a supertropical adjoint matrix

Abstract

Let $\chi (A)$ denote the characteristic polynomial of a matrix $A$ over a field; a standard result of linear algebra states that $\chi (A^{-1})$ is the reciprocal polynomial of $\chi (A)$. More formally, the condition ${\chi }^n(X){\chi }^k(X^{-1})={\chi }^{n-k}(X)$ holds for any invertible $n\times n$ matrix $X$ over a field, where ${\chi }^i(X)$ denotes the coefficient of ${\lambda }^{n-i}$ in the characteristic polynomial $\det (\lambda I-X)$. We confirm a recent conjecture of Niv by proving the tropical analogue of this result.

arXiv commentnote, 3 pages